Advertisement

On the Maxwell and Friedrichs/Poincaré constants in ND

  • Dirk PaulyEmail author
Article
  • 17 Downloads

Abstract

We prove that for bounded and convex domains in arbitrary dimensions, the Maxwell constants are bounded from below and above by Friedrichs’ and Poincaré’s constants, respectively. Especially, the second positive Maxwell eigenvalues in ND are bounded from below by the square root of the second Neumann-Laplace eigenvalue.

Keywords

Maxwell’s equations Maxwell constant Second Maxwell eigenvalue Electro statics Magneto statics Poincaré inequality Friedrichs inequality Poincaré constant Friedrichs constant 

Mathematics Subject Classification

35A23 35Q61 35E10 35F15 35R45 46E40 53A45 

Notes

Acknowledgements

We cordially thank the anonymous referee for a very careful reading and valuable suggestions for improving the paper.

References

  1. 1.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21(9), 823–864 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bao, G., Zhou, Z.: An inverse problem for scattering by a doubly periodic structure. Trans. Am. Math. Soc. 350(10), 4089–4103 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bauer, S., Pauly, D., Schomburg, M.: The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions. SIAM J. Math. Anal. 48(4), 2912–2943 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Filonov, N.: On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator. St. Petersburg Math. J. 16(2), 413–416 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations: theory and algorithms. Springer (Series in Computational Mathematics), Heidelberg (1986)CrossRefGoogle Scholar
  6. 6.
    Gol’dshtein, V., Mitrea, I., Mitrea, M.: Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds. J. Math. Sci. (N. Y.) 172(3), 347–400 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grisvard, P.: Elliptic problems in nonsmooth domains. Pitman (Advanced Publishing Program), Boston (1985)zbMATHGoogle Scholar
  8. 8.
    Jakab, T., Mitrea, I., Mitrea, M.: On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains. Indiana Univ. Math. J. 58(5), 2043–2071 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jochmann, F.: A compactness result for vector fields with divergence and curl in \({L}^q({\Omega })\) involving mixed boundary conditions. Appl. Anal. 66, 189–203 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kuhn, P., Pauly, D.: Regularity results for generalized electro-magnetic problems. Analysis (Munich) 30(3), 225–252 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Leis, R.: Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien. Math. Z. 106, 213–224 (1968)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leis, R.: Initial boundary value problems in mathematical physics. Teubner, Stuttgart (1986)CrossRefGoogle Scholar
  13. 13.
    Mitrea, M.: Dirichlet integrals and Gaffney-Friedrichs inequalities in convex domains. Forum Math. 13(4), 531–567 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pauly, D.: Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains. Adv. Math. Sci. Appl. 16(2), 591–622 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pauly, D.: Generalized electro-magneto statics in nonsmooth exterior domains. Analysis (Munich) 27(4), 425–464 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pauly, D.: Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains. Asymptot. Anal. 60(3–4), 125–184 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pauly, D.: Hodge-Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media. Math. Methods Appl. Sci. 31, 1509–1543 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pauly, D.: On constants in Maxwell inequalities for bounded and convex domains. Zapiski POMI 435, 46–54 (2014)Google Scholar
  19. 19.
    Pauly, D.: On constants in Maxwell inequalities for bounded and convex domains. J. Math. Sci. (N. Y.) 210(6), 787–792 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pauly, D.: On Maxwell’s and Poincaré’s constants. Discrete Contin. Dyn. Syst. Ser. S 8(3), 607–618 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pauly, D.: On the Maxwell constants in 3D. Math. Methods Appl. Sci. 40(2), 435–447 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Picard, R.: Randwertaufgaben der verallgemeinerten Potentialtheorie. Math. Methods Appl. Sci. 3, 218–228 (1981)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Picard, R.: On the boundary value problems of electro- and magnetostatics. Proc. R. Soc. Edinburgh Sect. A 92, 165–174 (1982)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Picard, R.: An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math. Z. 187, 151–164 (1984)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Picard, R.: Some decomposition theorems and their applications to non-linear potential theory and Hodge theory. Math. Methods Appl. Sci. 12, 35–53 (1990)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Picard, R., Weck, N., Witsch, K.-J.: Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles. Analysis (Munich) 21, 231–263 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Saranen, J.: Über das Verhalten der Lösungen der Maxwellschen Randwertaufgabe in Gebieten mit Kegelspitzen. Math. Methods Appl. Sci. 2(2), 235–250 (1980)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Saranen, J.: Über das Verhalten der Lösungen der Maxwellschen Randwertaufgabe in einigen nichtglatten Gebieten. Ann. Acad. Sci. Fenn. Ser. A I Math. 6(1), 15–28 (1981)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Saranen, J.: On an inequality of Friedrichs. Math. Scand. 51(2), 310–322 (1982)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Weber, C.: A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2, 12–25 (1980)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Weck, N.: Maxwell’s boundary value problems on Riemannian manifolds with nonsmooth boundaries. J. Math. Anal. Appl. 46, 410–437 (1974)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Witsch, K.-J.: A remark on a compactness result in electromagnetic theory. Math. Methods Appl. Sci. 16, 123–129 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

Personalised recommendations